Online Calibration of Path Loss Exponent in Wireless Sensor Networks

Transcription

1 Online Calibration of Path Loss Exponent in Wireless Sensor Networks Guoqiang Mao, Brian D.O. Anderson and Barış Fidan School of Electrical and Information Engineering, the University of Sydney. Research School of Information Sciences and Engineering, Australian National University National ICT Australia Limited[1], Sydney, Australia National ICT Australia Limited, Canberra, Australia Abstract The path loss exponent (PLE) is a parameter indicating the rate at which the received signal strength (RSS) decreases with distance, and its value depends on the specific propagation environment. Path loss exponent estimation plays an important role in distance-based wireless sensor network localization, where distance is estimated from the RSS measurements. Path loss exponent estimation is also useful for other purposes like sensor network dimensioning. Existing techniques on PLE estimation rely on both RSS measurements and distance measurements in the same environment to calibrate the PLE. However distance measurements can be difficult and expensive to obtain in some environments. In this paper we propose a technique for online calibration of the path loss exponent in wireless sensor networks without using distance measurements. The major contribution of this paper is to demonstrate that it is possible to estimate the PLE using only power measurements and the geometric constraints associated with planarity in a sensor network. This may have a significant impact on wireless sensor network localization. I. INTRODUCTION The wireless received signal strength (RSS) has been popularly modeled by a log-normal model [1], [], []: P ij [dbm] N(P ij [dbm], σ db), (1) P ij [dbm] = P (d )[dbm] α log (d ij /d ), () where P ij [dbm] is the received power at a receiving node j from a transmitting node i in db milliwatts, P ij [dbm] is the mean power in db milliwatts, σdb is the variance of the shadowing, P (d )[dbm] is the received power in db milliwatts at a reference distance d and d ij is the distance between nodes i and j. In this paper, we use the notation [dbm] to denote that power is measured in db milliwatts. Otherwise, it is measured in milliwatts. The reference power P (d )[dbm] is calculated using the free space Friis equation or obtained through field measurements at distance d []. It is important to notice that the reference distance d should always be in the far field of the transmitter antenna so that the near-field effect does not alter the reference power. In large coverage cellular systems, a 1 km reference distance is commonly used, whereas in microcellular systems, a much 1 National ICT Australia is funded by the Australian Governments Department of Communications, Information Technology and the Arts and the Australian Research Council through the Backing Australias Ability initiative and the ICT Centre of Excellence Program. smaller distance such as m or 1 m is used []. In this paper, it is assumed that d and P (d )[dbm] can be obtained from a priori calibration of the wireless device and they are known constants. In many cases, P ij [dbm] P ji [dbm] due to the asymmetric wireless signal propagation path from node i to node j and from node j to node i. However in this paper, it is assumed that P ij [dbm] = P ji [dbm] for simplicity. In the case of an asymmetric path, the average value of P ij [dbm] and P ji [dbm] is used. It is also assumed that all distances are normalized with respect to d. Therefore Eq. can be simplified as: P ij [dbm] = P (d )[dbm] αlog d ij. () The parameter α is called the path loss exponent (PLE), which indicates the rate at which the received signal strength decreases with distance. The value of α depends on the specific propagation environment. In this paper, it is assumed that α is an unknown constant, which is to be determined. For a large wireless sensor network spanning a wide area with different environmental conditions, the area can be divided into smaller regions and α can be considered as a constant in each region. Path loss exponent estimation plays an important role in distance-based wireless sensor network localization, where distance is estimated from the received signal strength measurements []. Path loss exponent estimation is also useful for other purposes like sensor network dimensioning. As the value of the path loss exponent depends on the environment in which a sensor network is deployed, existing techniques on path loss exponent estimation rely on both RSS measurements and distance measurements in the same environment to calibrate the path loss exponent [1], []. RSS measurements are readily available. However distance measurements can be difficult and expensive to obtain in some environments. Moreover, the reliance on distance measurements impedes deployment in unknown environments. In this paper we propose a technique for online calibration of the path loss exponent in wireless sensor networks using the Cayley-Menger determinant [], [], which does not rely on distance measurements. The proposed technique also does not assume knowledge of the quantity σdb in Eq. 1. The major contribution of this paper is to demonstrate that it is possible to estimate the PLE using only power measurements and the geometric constraints associated with planarity in a sensor

2 p 1 Fig. 1. P,d 1 1 P 1,d 1 P,d p p P,d 1 1 P,d P,d p A fully-connected planar quadrilateral in sensor network. network. This may have a significant impact on distance-based wireless sensor network localization. For ease of explanation, the paper focuses on path loss exponent estimation in D. However the proposed technique may also be extended to D. The rest of the paper is organized as follows. In Section II we present the fundamental principle underpinning the proposed technique as well as an introduction to the proposed algorithm. The presence of noise in the log-normal model means that considerable adjustment is required of an algorithm initially developed for the noiseless case, to correct for the presence of bias. In Section III, the proposed algorithm is validated using both simulations and real measurement data. Conclusions and suggestions to further work are given in Section IV. II. PATH LOSS EXPONENT ESTIMATION BASED ON THE CAYLEY-MENGER DETERMINANT Consider a sensor with three neighbors where the neighbors of that sensor are also neighbors of each other. Two sensors i and j are neighbors if P ij is nonzero. The sensor and its neighbors can be represented by a full-connected planar quadrilateral shown in Fig. 1. The tuple (P ij, d ij ) represents the measured power and the true distance between node i and node j respectively. P ij and d ij are related through Eq. 1 and Eq.. The path loss exponent estimation problem can be formulated as the simultaneous estimation of the true distances d 1, d 1, d 1, d, d, d and α given the power measurements P 1, P 1, P 1, P, P, P. Using the maximum likelihood estimator, the likelihood function can be obtained as: 1 i<j L(d 1, d 1, d 1, d, d, d, α) = 1 ( πσ db ) exp( (P ij[dbm] P (d )[dbm] + αlog d ij ) σdb ) As the number of power measurements is smaller than the number of parameters to be estimated, this leads to a set of simple equations. In these equations P ij and P (d ) are in decimal units, not in db units, so that P ij = Pij[dBm]/. P 1 = P (d ) d α 1 () P 1 = P (d ) d α 1 () P 1 = P (d ) d α 1 () P = P (d ) d α () P = P (d ) d α () P = P (d ) d α () Another constraint that is required to solve the above equations can be found from the geometric constraint on a fully-connected planar quadrilateral using the Cayley-Menger determinant [], []. The Cayley-Menger determinant of a quadrilateral is given by: D(p 1, p, p, p ) = d 1 d 1 d 1 1 d 1 d d 1 d 1 d d 1 d 1 d d () A classical result on the Cayley-Menger determinant is given by the following theorem: Theorem 1: (Theorem 11.1 in []) Consider an n-tuple of points p 1,..., p n in m-dimensional space with n m + 1. The rank of the Cayley-Menger matrix M(p 1,..., p n ) (defined analogously to the right side of Eq. but without the determinant operation) is at most m + 1. A direct application of the theorem leads to: D(p 1, p, p, p ) =. (11) Combining Eq. 11 and Eq. to Eq., a nonlinear equation for α can be obtained: C /α 1 C /α 1 C /α 1 1 C /α 1 C /α C /α 1 h(α) = C /α 1 C /α C /α 1 =, C /α 1 C /α C /α (1) where C ij = P ij /P (d ), 1 i < j, and the C ij are all known. An analytical solution to Eq. 1 is difficult to find. However Eq. 1 can be conveniently solved using the bracketing and bisection numerical technique [, section.1]. Unfortunately, the estimate of α obtained using this method shows a strong bias, i.e., Bˆα = E(ˆα) α. (1) Fig. shows a histogram of the estimated path loss exponent, which is obtained from simulation using, different quadrilaterals whose vertices are uniformly distributed in a square region of 1 1. Other parameters used in the simulation are: the true value of α is.; σ db =.; P (d )[dbm] =.dbm; d = 1m. These parameters are drawn from real measurement data reported in [1]. It should be noted that in the presence of noise in power measurements, Eq. 1 may have non-unique solutions even in the typical range of α

3 Number of estimated path loss exponent falling into the region Estimated path loss exponent Fig.. Histogram of the estimated path loss exponent using Cayley-Menger determinant. The figure is obtained using, quadrilaterals whose vertices are uniformly distributed in a square region of Fig.. Relationship between the bias of ˆα, the standard deviation of noise in power measurement σ db and α. B ˆα depends to a small degree on α as well as on σdb. for some quadrilaterals. In that case, we simply discard that quadrilateral and do not use it in the calculation to avoid any ambiguity. As shown in the figure, the true value of α is. and the estimated value of α has a bias of.. The simulation shows that ˆα obtained using the aforementioned technique has a strong bias which cannot be ignored. Bias removal requires an analysis on E( ˆα). However a direct analysis of E( ˆα) is difficult because of the difficulty in obtaining an explicit analytical expression of α from Eq. 1. Therefore we resort to numerical experiments to evaluate E(ˆα). Our numerical evaluation shows that the bias of ˆα, Bˆα, has an approximate linear relationship with σ db for a fixed α. This is shown in Fig.. Fig. is obtained using, quadrilaterals whose vertices are uniformly distributed in a square of 1 1. The corresponding power measurements are obtained using Eq. 1 and Eq.. Moreover further simulations show that the relationship between Bˆα, σ db and α is almost independent of the distribution of the vertices of the quadrilaterals and is also independent of the shape of the area in which the vertices of the quadrilaterals are located. As an example, Fig. shows the relationship between Bˆα and σ db for quadrilaterals whose vertices are located in areas of a variety of different shapes and distributed in the area following different distributions. Fig. shows the relationship between Bˆα and σ db at a specific value of α =.. Based on the observation shown in Fig. and Fig. that the relationship between Bˆα and σ db is independent of the distribution of the vertices of various quadrilaterals and the shape of the area in which vertices of the quadrilaterals are located, a pattern matching technique can be used to estimate the path loss exponent which uses the power measurements only. Specifically, via a priori simulation, a data base can be established ( where each entry in the ) data base is arranged in the form α i, σdb,j, E(ˆα) α i,σdb,j. The symbol E(ˆα) αi,σdb,j is used to emphasize dependence of E(ˆα) on the path loss exponent α i and the variance of noise in power measurements σdb,j. This data base can be obtained using a large number of quadrilterals whose vertices, say, are uniformly distributed in an area. The corresponding power measurements are obtained using Eq. 1 and Eq.. The distance between adjacent σdb is the same, i.e., σ db,j+1 σ db,j = σ db. (1) The distance between adjacent α is also the same, i.e., α i+1 α i = α. (1) Given this data base, the estimation of α can be obtained using the following procedure for pattern matching: 1) Identify a set of fully connected quadrilaterals in the wireless sensor network for further computation; ) Add a random Gaussian noise with variance σdb into each power measurement. The power is measured in db milliwatts unit; ) For each individual quadrilateral, solve for ˆα using Eq. 1. An E(ˆα) r,1 corresponding to σdb can be obtained as the average value of ˆα obtained from each individual quadrilateral. Here the subscript r is used to mark the difference with the corresponding value in the database; ) Repeat steps and using different noise variance values M σdb, where M =, 1,,..., m. Here M = corresponds to the original data set without the additionally introduced noise. A series of tuples (, E(ˆα) r, ), ( σdb, E(ˆα) r,1),..., (m σdb, E(ˆα) r,m) can then be obtained; ) Search the database and find the values of i and j such that: m {i, j} = argmin (E(ˆα) r,n E(ˆα) αi,σdb,j+n ). N= (1)

4 (a).. (b) (c) (d) Fig.. Relationship between the bias of α and the standard deviation of noise in power measurement σdb. The relationship between Bα and σdb is almost independent of the distribution of the vertices of the quadrilaterals and independent of the shape of the area in which the vertices of the quadrilaterals are located. a) Generated from quadrilaterals whose vertices are uniformly distributed in a L shaped area. b) Generated from quadrilaterals whose vertices are uniformly distributed in a rectangular area of x. c) Generated from quadrilaterals whose vertices are distributed in a square area of 1x1 following a truncated Gaussian distribution with a zero mean and a standard variation of. d) Generated from quadrilaterals whose vertices are located in a ring. The inner radius of the ring is and the outer radius is. The coordinates of the vertices are generated by first selecting a number uniformly distributed in [, ] and then selecting a number uniformly distributed in [, π]. The two numbers are used as the polar coordinate of the vertex to obtain the rectangular coordinate. The parameter m has to be a large number, say, in order to obtain a reasonably accurate estimate of α, which is robust against the randomness in the data. ) Finally, an improved estimate of α approximately correcting for the bias can be obtained as: α = αi. (1) Similarly, an estimate of σdb can be obtained as: σ db = σdb,j. (1) III. S IMULATION VALIDATION In this section, we shall validate the proposed technique using both simulations and real measurements. The database is established by simulations using, quadrilaterals whose vertices are uniformly distributed in a square region of 1 1. The measured power is generated using Eq. 1 and Eq.. σdb is set to be 1 and the distance between adjacent αi is set to be.1, i.e., α =.1. Another set of, quadrilaterals whose vertices is uniformly distributed in a rectangular area of are used to establish the performance of the proposed algorithm. points are used in the search, i.e., m = in Eq. 1. The simulation is repeated by varying the value of α from to and varying the value of σdb from to, which are the typical ranges of the two parameters []. Fig. shows the error in estimating α and Fig. shows the error in estimating σ. As shown in Fig., the error in estimating α is contained in the region [.,.1]. The mean estimation error is.11 and the error variance is.11. Fig. shows that the error in estimating σdb is contained in the region [, 1]. The mean estimation error is. and the error variance is.. Fig. also shows that the error in estimating σdb is larger at smaller values of σdb. This is attributable to the larger separation between σdb,j+1 and σdb,j at smaller values of σdb,j. The impact of the estimation error depends on the specific application, density of sensor network and deployment of sensors. It is beyond the scope of this paper to evaluate the impact of the estimation error. The estimation error is attributable to the randomness in the data and can be reduced by using smaller values of σdb and α at the expense of increased computational load. For example, our simulation (not shown here due to space limitation) shows that by using m =, α =. and

5 (a) (b) (c) (d) Fig.. Relationship between the bias of ˆα and the standard deviation of noise in power measurement σ db at a specific value of α. Subfigures a), b), c), d) are obtained under the same conditions as those in Fig.. σdb =., the maximum error in estimating α reduces to.. The mean error reduces to. and the error variance reduces to.. Another possible approach to reducing estimation error is by applying a noise reduction technique [] to data before the pattern matching. Simulations using quadrilaterals whose vertices are distributed in an area of a different shape and/or following a different distribution show similar performance. Fig. and Fig. show the simulation results using a set of, quadrilaterals whose vertices are distributed in a square region of 1 1 following a truncated two-dimensional Gaussian distribution. The mean of the Gaussian distribution is at the center of the square region and the standard deviation of the Gaussian distribution is. Both figures show similar performance as those in Fig. and Fig. except that the estimation error of α has a value of. at a couple of points in Fig.. However the estimation error of σ db is better than that in Fig.. To further evaluate the performance of the proposed technique, we apply it to real measurement data reported in [1]. The measurement data and the deployment of sensors can also be found at hero/localize/. The wireless sensor network in [1] consists of fully connected nodes, which make up 1, 1 quadrilaterals. It is both computationally intensive and unnecessary to compute ˆα for each quadrilateral. Here we randomly choose, quadri- Error in Estimating α Fig.. Error in estimating α using the pattern matching. The vertices of the quadrilaterals are uniformly distributed in a rectangular area of. laterals for computation. The reference power P (d )[dbm] is calculated using the free space Friis equation at a reference distance d = 1m and P (d )[dbm] =.dbm [1]. The path loss exponent estimated using both the power measurements and the measured distances is.. The path loss exponent estimated using the proposed technique is., which represents an error of.. The true value σ db

6 Error in Estimating σ Fig.. Error in estimating σ db using the pattern matching. The vertices of the quadrilaterals are uniformly distributed in a rectangular area of. Error in Estimating α Fig.. Error in estimating α using the pattern matching. The vertices of the quadrilaterals are distributed in a square region of 1 1 following a truncated two-dimensional Gaussian distribution. (i.e., the value obtained using real power measurements and measured distances) is. and the estimation error is 1.. The slightly larger error in estimating σ db when using real measurement data may be attributable to the density of the noise in the power measurements deviating from a Gaussian distribution. IV. CONCLUSION AND FURTHER WORK In this paper, we proposed an algorithm which can estimate the path loss exponent using only power measurements and the underlying planar geometric constraints on the sensors only. The algorithm is based on the Cayley-Menger determinant and it does not use any distance measurements. The algorithm is validated using both simulations and real measurement data. The proposed algorithm may have significant impact on distance-based wireless sensor network localization. In this paper, we observed the empirical law that the relationship between Bˆα, σ db and α is independent of the distribution of the vertices of the quadrilaterals and is also independent of.. Error in Estimating σ Fig.. Error in estimating σ db using the pattern matching. The vertices of the quadrilaterals are distributed in a square region of 1 1 following a truncated two-dimensional Gaussian distribution. the shape of the area in which the vertices of the quadrilaterals are located. It is desirable to obtain an analytical expression of the relationship between Bˆα, σ db and α. This is also the direction of our future research. Furthermore, the proposed algorithm relies on the lognormal propagation model in Eq. 1 and Eq. in the sense that the maximum likelihood estimator shown in Eq. to Eq. may have a different form when the received signal strength has a different model. Although the log-normal propagation model is a popular model for wireless network, there are environments in which the log-normal propagation model is not the best model []. In that case, a technique needs to be developed to select the best model and choose the best estimator for distance to replace Eq. to Eq. accordingly. Therefore how to develop an algorithm for environments in which the lognormal propagation model does not apply is also a future research topic. REFERENCES [1] N. Patwari, I. Hero, A.O., M. Perkins, N. Correal, and R. O Dea, Relative location estimation in wireless sensor networks, IEEE Transactions on Signal Processing, vol. 1, no., pp. 1 1,. [] T. S. Rappaport, Wireless Communications: Principles and Practice, nd ed. Prentice Hall PTR, 1. [] D. Lymberopoulos, Q. Lindsey, and A. Savvides, An empirical analysis of radio signal strength variability in ieee.1. networks using monopole antennas, Tech. Rep. ENALAB Technical Report 1,. [] G. Mao, B. Fidan, and B. D. O. Anderson, Sensor network localization, in Sensor Network and Configuration: Fundamentals, Techniques, Platforms and Experiments. Germany: Springer-Verlag,. [] G. M. Crippen and T. F. Havel, Distance Geometry and Molecular Conformation. New York: John Wiley and Sons Inc., 1. [] L. M. Blumenthal, Theory and Applications Distance Geometry. Oxford University Press, 1. [] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Receipes in C - The Art of Scientific Computing, nd ed. Cambridge University Press,. [] S. V. Vaseghi, Advanced Digital Signal Processing and Noise Reduction, rd ed. John Wiley & Sons,...